Hostname: page-component-84b7d79bbc-tsvsl Total loading time: 0 Render date: 2024-07-31T09:59:22.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings Characterized via a Class of Left Exact Preradicals

Part of: Rings and algebras arising under various constructions Radicals and radical properties of rings Modules, bimodules and ideals

Published online by Cambridge University Press:  17 December 2015

Noyan Er
Noyan Er*
Affiliation:
Yıldız Technical University, Department of Mathematical Engineering, Istanbul, Turkey (noyaner@yahoo.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

For two modules M and N, iM(N) stands for the largest submodule of N relative to which M is injective. For any module M, iM: Mod-R → Mod-R thus defines a left exact preradical, and iM(M) is quasi-injective. Classes of ring including strongly prime, semi-Artinian rings and those with no middle class are characterized using this functor: a ring R is semi-simple or right strongly prime if and only if for any right R-module M, iM(R) = R or 0, extending a result of Rubin; R is a right QI-ring if and only if R has the ascending chain condition (a.c.c.) on essential right ideals and iM is a radical for each M ∈ Mod-R (the a.c.c. is not redundant), extending a partial answer of Dauns and Zhou to a long-standing open problem. Also discussed are rings close to those with no middle class.

Keywords

Ringpreradicalinjective

MSC classification

Secondary: 16D50: Injective modules, self-injective rings 16N60: Prime and semiprime rings 16S90: Torsion theories; radicals on module categories
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 641 - 653
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Alahmadi, A. N., Alkan, M. and López-Permouth, S. R., Poor modules: the opposite of injectivity, Glasgow Math. J. 52 (2010), 7–17.CrossRefGoogle Scholar
2. Anderson, F. and Fuller, K., Rings and categories of modules, Graduate Texts in Mathematics, Volume 13 (Springer, 1992).CrossRefGoogle Scholar
3. Bican, L., Kepka, T. and Nemec, P., Rings, modules, and preradicals, Lecture Notes in Pure and Applied Mathematics, Volume 75 (Dekker, New York, 1982).Google Scholar
4. Boyle, A., Hereditary QI-rings, Trans. Am. Math. Soc. 192 (1974), 115–120.Google Scholar
5. Dauns, J. and Zhou, Y., Sublattices of the lattice of pre-natural classes of modules, J. Alg. 231 (2000), 138–162.Google Scholar
6. Er, N., López-Permouth, N. S. R. and Sökmez, N., Rings whose modules have maximal or minimal injectivity domains, J. Alg. 330(1) (2011), 404–417.CrossRefGoogle Scholar
7. Goldman, O., Rings and modules of quotients, J. Alg. 13 (1969), 10–47.CrossRefGoogle Scholar
8. Goodearl, K., Singular torsion and the splitting properties, Memoirs of the American Mathematical Society, Volume 124 (American Mathematical Society, Providence, RI, 1972).Google Scholar
9. Goodearl, K., Handelman, D. and Lawrence, J., Strongly prime and completely torsion-free rings, Carleton Mathematical Society (Carleton University Press, Ottawa, ON, 1974).Google Scholar
10. Handelman, D., Strongly semiprime rings, Pac. J. Math. 60(1) (1975), 115–122.CrossRefGoogle Scholar
11. Handelman, D. and Lawrence, J., Strongly prime rings, Trans. Am. Math. Soc. 211 (1975), 209–223.CrossRefGoogle Scholar
12. López-Permouth, S. R. and Simental, J. E., Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Alg. 362 (2012), 56–69.Google Scholar
13. Mohamed, S. H. and Müller, B. J., Continuous and discrete modules, London Mathematical Society Lecture Note Series, Volume 147 (Cambridge University Press, 1990).CrossRefGoogle Scholar
14. Rubin, R., Absolutely torsion-free rings, Pac. J. Math. 46 (1973), 503–514.CrossRefGoogle Scholar
15. Teply, M. L., On the idempotence and stability of kernel functors, Glasgow J. Math. 37 (1995), 37–43.CrossRefGoogle Scholar
16. van den Berg, J. E., When every torsion preradical is a torsion radical, Commun. Alg. 27(11) (1999), 5527–5547.CrossRefGoogle Scholar
17. Viola-Prioli, J., On absolutely torsion-free rings and kernel functors, PhD Thesis, Rutgers University (1973).Google Scholar